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A198954 Expansion of the rotational partition function for a heteronuclear diatomic molecule. 3
1, 3, 0, 5, 0, 0, 7, 0, 0, 0, 9, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The partition function of a heteronuclear diatomic molecule is Sum_{J>=0} (2*J + 1) * exp( - J * (J + 1) * hbar^2 / (2 * I * k * T)) where I is the moment of inertia, hbar is reduced Planck's constant, k is Boltzmann's constant, and T is temperature. The degeneracy for the J-th energy level is 2*J + 1.

As triangle : triangle T(n,k), read by rows, given by (3,-4/3,1/3,0,0,0,0,0,0,0,...) DELTA (0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - From Philippe Deléham, Nov 01 2011

Note that the g.f. theta_1'(0, q^(1/2)) / (2 * q^(1/8)) = 1 - 3*q  + 5*q^3 - 7*q^6 + 9*q^10 + ... which is the same as this sequence except the signs alternate. - Michael Somos, Aug 26 2015

REFERENCES

G. H. Wannier, Statistical Physics, Dover Publications, 1987, see p. 215 equ. (11.13).

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10011

FORMULA

G.f.: Sum_{k>=0} (2*k + 1) * x^( (k^2 + k) / 2). This is related to Jacobi theta functions.

a(n) = (t*(t+1)-2*n-1)*(t-r), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - Mikael Aaltonen, Jan 15 2015

a(n) = A053187(2n+1) - A053187(2n). - Robert Israel, Jan 15 2015

a(n) = abs(A010816(n)). - Joerg Arndt, Jan 16 2015

EXAMPLE

G.f. = 1 + 3*x + 5*x^3 + 7*x^6 + 9*x^10 + 11*x^15 + 13*x^21 + 15*x^28 + ...

G.f. = 1 + 3*q^2 + 5*q^6 + 7*q^12 + 9*q^20 + 11*q^30 + 13*q^42 + 15*q^56 + ...

Triangle begins :

1

3, 0

5, 0, 0

7, 0, 0, 0

9, 0, 0, 0, 0

11, 0, 0, 0, 0, 0

13, 0, 0, 0, 0, 0, 0

15, 0, 0, 0, 0, 0, 0, 0

17, 0, 0, 0, 0, 0, 0, 0, 0

MAPLE

seq(op([2*i+1, 0$i]), i=0..10); # Robert Israel, Jan 15 2015

MATHEMATICA

a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ[m], m KroneckerSymbol[ 4, m], 0]]]; (* Michael Somos, Aug 26 2015 *)

PROG

(PARI) {a(n) = my(m); if( issquare( 8*n + 1, &m), m, 0)};

CROSSREFS

Cf. A053187, A107270.

Sequence in context: A154725 A010816 A133089 * A136599 A227498 A131986

Adjacent sequences:  A198951 A198952 A198953 * A198955 A198956 A198957

KEYWORD

nonn,tabl,easy

AUTHOR

Michael Somos, Oct 31 2011

STATUS

approved

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Last modified January 20 11:06 EST 2020. Contains 331083 sequences. (Running on oeis4.)